The construction of bounded game-theoretic controls for non-linear dynamical systems

Abstract

The problem of transferring a non-linear dynamical system, subject to perturbations, to the null equilibrium position in a finite time by means of a bounded control is considered. Only the levels of uncontrollable perturbations are known, and are not assumed to be small. Sufficient conditions are obtained which ensure that the problem has a guaranteed solution for the given domain of initial conditions. An estimate of the guaranteed control time is obtained. The construction of the control can be reduced to the construction of game strategies for auxiliary linear game-theoretic problems. To estimate the “auxiliary noise” in the resulting linear system, the principle of “prescribing and subsequent confirmation” of noise levels is put forward. On the basis of this principle, these estimates are checked on the set of states of the auxiliary linear systems, where the control is also subsequently estimated. As a result, an iterative algorithm for solving the original non-linear problem is obtained. Within the framework of the method proposed a new solution of the game-theoretic problem of the reorientation of an asymmetric rigid body in the presence of noise is given.